Sample QuestionsModel Paper 6 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular to each other then $k =$ ?
- ✓
$\frac{-10}{7}$
- B
$\frac{5}{7}$
- C
$\frac{-5}{7}$
- D
$\frac{10}{7}$
Answer: A.
View full solution →If $x = a \sec \theta, y = b \tan \theta$ then $\frac{d y}{d x}=$ ?
- A
$\frac{b}{a} \sec \theta$
- B
$\frac{b}{a} \tan \theta$
- ✓
$\frac{b}{a} \operatorname{cosec} \theta$
- D
$\frac{b}{a} \cot \theta$
Answer: C.
View full solution →If $|\vec{a}|=4$ and $-3 \leq \lambda \leq 2$, then the range of $|\lambda \vec{a}|$ is
- ✓
$[0,12]$
- B
$[0,8]$
- C
$[8,12]$
- D
$[-12,8]$
Answer: A.
View full solution →The general solution of a differential equation of the type $\frac{d x}{d y}+ P _1 x= Q _1$ is
- ✓
$x e^{\int P _1 d y}=\int\left( Q _1 e^{\int P _1 d y}\right) d y+ C$
- B
$y e^{\int P _1 d y}=\int\left( Q _1 e^{\int P _1 d y}\right) d y+ C$
- C
$y \cdot e^{\int_P d x}=\int\left(Q_1 e^{\int P_1 d x}\right) d x+C$
- D
$x e^{\int P^1 d x}=\int\left(Q_1 e^{\int P_1 d x}\right) d x+C$
Answer: A.
View full solution →$X$ and $Y$ are independent events such that $P(X \cap \bar{Y})=\frac{2}{5}$ and $P(X)=\frac{3}{5}$. Then $P(Y)$ is equal to:
- A
$\frac{2}{3}$
- ✓
$\frac{1}{3}$
- C
$\frac{1}{5}$
- D
$\frac{2}{5}$
Answer: B.
View full solution →Assertion (A): Let $A=\{2,4,6\}$ and $B=\{3,5,7,9\}$ and defined a function $f=\{(2,3),(4,5),(6,7)\}$ from $A$ to
B. Then, f is not onto.
Reason (R): A function $f$ : $A \rightarrow B$ is said to be onto, if every element of $B$ is the image of some elements of $A$ under $f$.
- ✓
Both A and R are true and R is the correct explanation of A.
- B
Both A and R are true but R is not the correct explanation of A.
- C
A is true but R is false.
- D
A is false but R is true.
Answer: A.
View full solution →Assertion $(A)$ : If two positive numbers are such that sum is $16$ and sum of their cubes is minimum, then numbers are $8, 8.$
Reason $(R)$ : If f be a function defined on an interval $I$ and $c \in l$ and let $f$ be twice differentiable at $c,$ then $x = c$ is a point of local minima if $f\ '(c) = 0$ and $f\ "(c) > 0 $ and $f(c)$ is local minimum value of $f$.
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Show that $f(x)=(x-1) e^x+1$ is an increasing function for all $x>0$.
View full solution →Prove that: $\int_0^{\pi / 2} \frac{d x}{(1+\sqrt{\tan x})}=\frac{\pi}{4}$
View full solution →The volume of a sphere is increasing at the rate of $8 \ cm^3 / s$. Find the rate at which its surface area is increasing when the radius of the sphere is $12 \ cm .$
View full solution →Find the intervals in which the function $f$ given by $f(x)=2 x^3-3 x^2-36 x+7$ is decreasing.
View full solution →A man is walking at the rate of $6.5 \ km/hr$ towards the foot of a tower $120 m$ high. At what rate is he approaching the top of the tower when he is $50 m$ away from the tower
View full solution →If $e ^{ x }+ e ^{ y }= e ^{ x + y }$, prove that $\frac{d y}{d x}+ e ^{ y - x }=0$.
View full solution →The corner points of the feasible region determined by the system of linear inequations are as shown below:
Answer each of the following:
$i.$ Let $z = 13x - 15y$ be the objective function. Find the maximum and minimum values of $z$ and also the
corresponding points at which the maximum and minimum values occur.
$ii.$ Let $z = kx + y$ be the objective function. Find $k,$ if the value of $z$ at $A$ is same as the value of $z$ at $B.$ View full solution →Solve the following $\text{LPP}$ graphically: Minimise $Z=5 x+10 y$ subject to the constraints $x+2 y \geq 120 \ x+y \geq 60, x-2 y \geq 0 $ and $x, y \geq 0$
View full solution →Find a particular solution of the differential equation $\frac{d y}{d x}+2 y \tan x=\sin x$, given that $y =0$, when $x=\frac{\pi}{3}$.
View full solution →$\left(x^2+y^2\right) d y=x y d x$. If $y(1)=1$ and $y\left(x_0\right)=e$, then find the value of $x_0$.
View full solution →Find the perpendicular distance of the point $(1,0,0)$ from the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$. Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.
View full solution →Find the shortest distance between the given lines. $\vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$, $\vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k})$
View full solution →Given $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1\end{array}\right]$ find $AB$ and use this result in solving the following system of equations. $x - y + z =4 , x - 2y - 2z = 9 , 2x + y + 3z = 1$
View full solution →Show that the function $f : R _0 \rightarrow R _0$, defined as $f ( x )=\frac{1}{x}$, is one-one onto, where $R _0$ is the set non-zero real numbers.
Is the result true, if the domain $R _0$ is replaced by N with co-domain being same as $R _0$ ?
View full solution →Let $R$ be relation defined on the set of natural number $N$ as follows:
$R=\{(x, y): x \in N, y \in N, 2 x+y=41\}$ Find the domain and range of the relation $R.$ Also verify whether $R$ is reflexive, symmetric and transitive.
View full solution →Read the following text carefully and answer the questions that follow:
An Apache helicopter of the enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $(3,7)$ want to shoot down the helicopter when it is nearest to him.
$i.$ If $P \left( x _1, y _1\right)$ be the position of a helicopter on curve $y=x^2+7$ then find distance $D$ from $P$ to soldier place at $(3.7).(1)$
$ii.$ Find the critical point such that distance is minimum. $(1)$
$iii$. Verify by second derivative test that distance is minimum at $(1, 8). (2)$
OR
Find the minimum distance between soldier and helicopter? $(2)$ View full solution →Read the following text carefully and answer the questions that follow:
Three friends Ganesh, Dinesh and Ramesh went for playing a Tug of war game. Team $A, B,$ and $C$ belong to Ganesh, Dinesh and Ramesh respectively.
Teams $A, B, C$ have attached a rope to a metal ring and is trying to pull the ring into their own area $($team areas shown below$)$.
Team $A$ pulls with $F _1=4 \hat{i}+0 \hat{j} KN$
Team $B \rightarrow F _2=-2 \hat{i}+4 \hat{j} KN$
Team $C \rightarrow F _3=-3 \hat{i}-3 \hat{j} KN$
$i$. Which team will win the game? $(1)$
$ii$. What is the magnitude of the teams combine Force? $(1)$
$iii.$ What is the magnitude of the force of Team $B? (2)$
OR
How many $KN$ Force is applied by Team $A? (2)$ View full solution →Read the following text carefully and answer the questions that follow:
For an audition of a reality singing competition, interested candidates were asked to apply under one of the two musical genres$-$folk or classical and under one of the two age categories$-$below $18$ or $18$ and above.
The following information is known about the $2000$ application received:
$i. 960$ of the total applications were the folk genre.
$ii. 192$ of the folk applications were for the below $18$ category.
$iii. 104$ of the classical applications were for the $18$ and above category.
Questions:
$i.$ What is the probability that an application selected at random is for the $18$ and above category provided it is under the classical genre? Show your work. $(1)$
$ii.$ An application selected at random is found to be under the below $18$ category. Find the probability that it is under the folk genre. Show your work. $(1)$
$iii.$ If $P(A)=0.4, P(B)=0.8$ and $P(B \mid A)=0.6$, then $P(A \cup B)$ is equal to. $(2)$
$OR$
$iv.$ If $A$ and $B$ are two independent events with $P ( A )=\frac{3}{5}$ and $P ( B )=\frac{4}{9}$, then find $P \left( A ^{\prime} \cap B ^{\prime}\right). (2)$
View full solution →